University of Nebraska - Lincoln
DigitalCommons@University of Nebraska - Lincoln
John R. Hardy Papers
Research Papers in Physics and Astronomy
5-15-1971
Calculation of Point-Defect Energies and
Displacements in Alkali Halides Using the LatticeStatics Method
Arnold Karo
University of California, Livermore, California
John Hardy
University of Nebraska - Lincoln
Follow this and additional works at: http://digitalcommons.unl.edu/physicshardy
Part of the Physics Commons
Karo, Arnold and Hardy, John, "Calculation of Point-Defect Energies and Displacements in Alkali Halides Using the Lattice-Statics
Method" (1971). John R. Hardy Papers. Paper 21.
http://digitalcommons.unl.edu/physicshardy/21
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska Lincoln. It has been accepted for inclusion in John R. Hardy Papers by an authorized administrator of DigitalCommons@University of Nebraska Lincoln.
PHYSICAL REVIEW B
VOLUME 3, NUMBER 10
15 MAY 1971
Calculation of Point-Defect Energies and Displacements in
Alkali Halides Using the Lattice-Statics ~ e t h o d *
Arnold M. Karo
Lawrence Radiation Laboratory, University of California, L i v e r m o r e , Ca1i;fornia 94550
and
John R. Hardy
Behlen Laboratory of P h y s i c s , University of Nebraska, Lincoln, Nebraska 68508
(Received 27 March 1970)
The method of lattice statics, in the zero-order approximation, i s used to evaluate the
Schottky-pair formation energies and the displacement fields about positive- and negative-ion
vacancies f o r all alkali halide crystals with rock-salt structures. The deformation-dipole
model, used extensively in e a r l i e r work on the lattice dynamics of these crystals, i s employed to describe the host lattice. Long-range Coulomb interactions a r e included explicitly, and short-range repulsive interactions a r e described by a simple Born-Mayer potential.
Comparison i s made with e a r l i e r Mott-Littleton-type calculations and very satisfactory
agreement i s found. Comparison with available experimental values i s somewhat l e s s satisfactory, particularly regarding the lithium salts; but for "typical" alkali halides, e. g. ,
NaCl and KC1, the calculated values a r e within 10% of the experimental values.
I. INTRODUCTION
In 1938, Mott and ~ i t t l e t o n 'presented their class i c calculation of the formation energies of Schottky
pairs in ionic crystals. The Schottky pair i s defined a s a pair of positive- and negative-ion vacancies, and the formation energy is that energy r e quired to remove the ions from the vacated lattice
sites and place them on the crystal surface.
The Mott-Littleton and subsequent calculations
have been based on the assumption that the crystal
energy can be written a s a sum of pairwise t e r m s .
In the simplest case, the lattice energy i s considered to be composed of the Coulomb energy of the
positive and negative ions and the energy associated
with a short-range repulsive interaction between
nearest neighbors. The latter can be said to a r i s e
from overlap of the electron densities of neighboring ions a s these a r e brought together in the crystal. The increase in local electron density with
decreasing internuclear separation results in increasing Coulomb and exchange contributions that
stabilize the lattice. Calculations by ~Bwdin'support such a simple model although indicating the
need for considering three- and possibly four-body
t e r m s in lattice-energy calculations.
As Mott and Littleton have shown, the energy r e quired to create a Schottky pair i s not simply that
needed to remove the ions from their lattice sites
and place them on the surface. If this were the
case, the formation energy would be the lattice
energy per ion pair. However, the lattice energy
per pair i s typically about 8 eV, whereas observed
formation energies a r e about 2 eV. The difference
a r i s e s because the lattice about a defect does not
remain undistorted and unpolarized. One thus inf e r s that the relaxation energy, i. e . , the energy
recovered when the lattice polarizes and distorts,
is of the order of 6 eV.
In any theoretical computation, the formation
energy appears a s the difference between two larger
quantities-the lattice energy and the relaxation energy-and care in evaluating these quantities i s
therefore important. Since the lattice energy can
be precisely computed without difficulty, it i s the
relaxation energy that presents the problem, and
the Mott-Littleton technique has been used to date
to calculate this value. Such defect calculations
have been characterized a s "semidiscrete" : In
such calculations the lattice is divided into two
regions labeled I and 11. The first region, I , i s
treated on a discrete atomistic basis, while the
second region, 11, i s represented by a continuum
approximation. In simple Mott-Littleton calculations region I includes only the defect and i t s nearest neighbors and the second region i s considered
a s apolarizeddielectric continuum. Recent calculations by Scholz4 have extended region I t o include
more than 200 ions, region II again being represented by a dielectric continuum. In a l l these calculations there is the difficulty of matching regions
I and I1 at the boundaries. For defects in metals it
has been foundS that Mott-Littleton calculations
would require region I to be inordinately large for
the boundaries to be in the true continuum region.
For ionic crystals, if the defect is charged, there
i s the additional complication of allowing for the
long-range forces exerted by the defect on the ions
of the host lattice. These forces cause the ions to
displace and polarize, creating additional problems
3
CALCULATION O F POINT-DEFECT ENERGIES
in the matching of regions I and 11.
In an earlier paper5 the formalism for the method
of lattice statics was presented, and it was shown
that the formation energies of charged point defects
in ionic crystals could be calculated without the
artificial dichotomy between regions I and 11. In
that paper the principal concern was the method of
calculating the displacements and polarizations and
establishing their asymptotic behavior far from the
defect. Our purpose in this paper is to present the
results of numerical calculations for the rock-saltstructure alkali halide sequence based onthe latticestatics method and with assumptions about ionic interactions similar to those made by Mott and Littleton. We shall restrict the calculations to the formation energies of Schottky p a i r s ; however, the
proposed method is general and can be applied
equally well to other point defects such a s substitutional impurities and interstitials.
11. THEORY
In this paper, we shall not repeat in detail the
formal theory given in full in Ref. 5. It is worth
emphasizing, however, that the crucial step in all
these calculations is the Fourier transformation of
the equilibrium equations for displacements and
polarizations of the ions. It should be noted also
that the ionic dipole moments must appear a s e s sential additional degrees of freedom having their
own equilibrium equation since, for a charged defect, a generalized force is applied to these electronic dipole moments a s well a s to the displacements. The model used here to describe the lattice
is the deformation-dipole model also discussed in
detail in our earlier paper.6 This model is a generalization of ideas originally developed by szigeti7
for treating long-wavelength lattice vibrations or
periodic distortions of arbitrary wavelength. Because of the finite number of ions in a lattice, it
follows that there a r e but a finite number of allowed
distinct wave vectors. These vectors a r e such
that the number of Fourier components of the displacements, o r of the dipole moments, i s exactly
equal to the total number of degrees of freedom of
the lattice a s a whole. From the detailed derivation of the lattice-statics method in Ref. 5, we can
extract the following basic equations [i. e., the definition of the energy function W and Eqs. (2.16) and
(2. 17) of Ref. 51 on which the present calculations
a r e based :
w = x c z : I , i;.)+r(T,
Z) ,
(1)
ymin=$ [ E ( S U +1 ) ~ " ; , ~ , + B ~ , ~ , , + E ~ , ~,, ] (2)
Ymi,=${[ ? + E c - ' ( I + S U ) U - ' ]
x [ ~ - ' ( 1 +U S ) ~ " E+
in which
and
M-I
1/11+ $,$c-lcuE ,
(3)
denote the displacements and mo-
...
3419
ments in the region where the potential-energy function i s harmonic. In the first equation, the total
energy of the solid is expressed a s the sum of two
terms. The first t e r m represents the interaction
between the defect and the ions of the host lattice,
and the second t e r m represents the energy stored
in the distorted and polarized but otherwise perfect
lattice. The distortion is assumed to be within the
limits of the harmonic approximation. The total
energy Wof the solid is minimized with respect to
both the displacements and the dipole moments, and
both quantities a r e then eliminated by expressing
them in t e r m s of the force exerted by the defect.
As a result we have Eq. (2), from which Eq. (3)
follows by substituting the appropriate expressions
for
and
The matrices have been defined
in Ref. 5, and the quantity Yminin Eq. (3) is the
energy stored in the distorted perfect lattice. If
the first t e r m in Eq. (3) is expanded a s a power
s e r i e s in the nuclear displacements and dipole moments, it can be shown5 that the change in energy
of the crystal when the lattice is allowed t o relax
and polarize (5e. , the relaxation energy ER) is, to
first order in tminand ,&,in, given by
Tmin
The expression for Y,,, is written entirely in t e r m s
of the forces, both Coulomb and repulsive, exerted
by the defect on the lattice ions at their unrelaxed
positions. We refer to this a s the zero-order approximation, and the calculations in the present
paper a r e made within this approximation.
The evaluation of ER i s accomplished by Fouriertransforming Eq. (3) and obtaining a sum over the
allowed wave vectors, given in the following form:
- $C3[ ? +Ec-'(I + S U)U-'I 111-'
ER=
X [ U " ( ~ + U S ) ~ - ' E + V ] - $(5)
C~~C-~~E,
in which the direct-space vectors and matrices a r e
replaced by their Fourier transforms. Two important points should be emphasized. First, at no
stage in this calculation has a distinction been made
between region I and region 11; the expression for
Y,,, refers to the lattice a s a whole. Second, the
long-range Coulomb forces, after Fourier transformation, can be handled by the Ewald 6-function
transformatione in the manner described in Appendix I of Ref. 5. The zero-order approximation is
particularly convenient because the actual displacements and dipole moments need not be introduced
o r evaluated in treating these long-range forces.
To proceed beyond this approximation, techniques
must be followed similar to those used in treating
metals by the lattice-statics m e t h ~ dwhere
,~
the
displacements of close neighbors of the defect were
explicitly considered-these being the largest and
also possibly outside the limits of the harmonic
approximation.
3420
A. M . K A R O A N D J. R . H A R D Y
3
111. RESULTS AND DISCUSSION
It is relevant to summarize results of a survey of
the alkali halide Schottky-pair formation energies
and associated displacement fields about the vacancies. The energy calculations have been carried
out within the zero-order approximation, and the
results can be compared appropriately with MottLittleton-type calculations and with experimental
data. In Table I we collect the formation energies
of Schottky pairs evaluated in this way. Listed
separately a r e the energies required to remove the
positive and negative ions to infinity (E + and E.),
the lattice energy per unit cell (E,), and the formation energy of the Schottky pair ( E s = E + + E- +EL).
Implied is an assumption that the energy of the perfect crystal can be written a s a sum of pairwise
terms. Thus, for the lattice energies we have a
simple Born-Mayer form
-
~1
Q,
w
rl
w
l
i
I:
9
-g.
m
ri
';[
z z
i
3
PI
<I-;,
.$
$i
3
8
"
6
u
met4
restricting the short-range interaction to nearestneighbor ions. The constants A and p a r e determined in the usual way from the lattice-equilibrium
condition and the observed compressibility. Taken
together, these define the first and second derivatives of the short-range interaction and thereby A
and p . The matrix M i s essentially the dynamical
matrix, without pre- and postmultiplication by the
diagonal mass matrix, defined for the deformationdipole model.'
Tables I1 and 111 summarize our calculations of
the displacements of neighboring ions about the defects and along several symmetry directions for
each alkali halide. Only representative results a r e
given for the symmetry directions in Table 111.
Complete data a r e available upon requesteg The
calculations follow from the expression
M J = U - ~ (+I ~
~
1
+ v- ,l
t
~
(7)
or, by the Fourier transform
M Q = u-'(I+ US)C-IE +
v,
(8)
which a r e Eqs. (2. 14) and (3.1) of Ref. 5. One
thenback-transforms,i. e., sums the Fourier s e r i e s
for the direct-space displacement of each ion. We
have, in fact, been somewhat more sophisticated
in calculating the displacements versus the relaxation energy, since the short-range force V is evaluated at the first-neighbor relaxed position. In
doing this, we note that $valuation of the firstneighbor displacements t x nabout the defect involves Eq. ('7)) which is nonlinear in these displacements since the short-range force i s itself a
function of the first-neighbor displacement. The
evaluation can be done iteratively and, once accomplished, yields the appropriate form of V for
use in calculating the relaxation energy of any
neighbor. Stated more precisely, we know the ap-
TABLE II. Displacement components f o r certain near neighbors about a positive-ion vacancy (+) o r negative-ion vacancy (-). Values a r e given in percent of the nearneighbor distance in the perfect lattice. A positive value indicates outward relaxation.
Neighbor
=I
L 2 -53
(+)
tx
t y
(-)
t;e
5x
tY
(+)
[e
t x
[Y
(-)
5,
Li F
1
1
1
2
1
1
2
2
3
2
0
1
1
0
2
1
2
2
0
2
0
0
1
0
0
2
0
1
0
2
12.356
-1.635
0.160
-0.622
1.752
-0.567
-0.377
0.345
0.589
- 0.316
0.0
0.160
0.0
1.654
-0.567
-0.377
0.345
0.0
-0.316
0.0
0.0
0.160
0.0
0.0
-0.455
0.0
0.444
0.0
-0.316
1
1
1
2
1
1
2
2
3
2
0
1
1
0
2
1
2
2
0
2
0
0
1
0
0
2
0
1
0
2
5.433
-0.798
0.331
-3.134
1.225
-0.243
-0.112
0.268
-0.767
- 0.078
0.0
-0.798
0.331
0.0
0.779
- 0.243
- 0.112
0.268
0.0
-0.078
0.0
0.0
0.331
0.0
0.0
- 0.129
0.0
0.352
0.0
-0.078
1
1
1
2
1
1
2
2
3
2
0
1
1
0
2
1
2
2
0
2
0
0
1
0
0
2
0
1
0
2
11.128
- 1.965
0.441
1.012
1.191
-0.600
-0.415
0.305
1.435
-0.347
0.0
-1.965
0.441
0.0
1.286
-0.600
-0.415
0.305
0.0
-0.347
0.0
0.0
0.441
0.0
0.0
-0.677
0.0
0.277
0.0
-0.347
- 1.635
t x
[Y
5e
LiCl
- 1.316
16.057
- 1.316
0.0
0.186
1.428
2.227
- 0.432
0.024
0.561
0.454
-0.124
0.186
0.0
2.355
- 0.432
0. 024
0.561
0.0
-0.124
0.0
0.0
0.186
0.0
0.0
- 0.432
0.0
0.609
0.0
-0.124
- 0.925
13.839
- 1.274
0.263
3.861
1.232
- 0.385
0.045
0.264
3.028
- 0.170
0.0
- 1.274
0.263
0.0
1.359
- 0.385
0.045
0.264
0.0
- 0.170
13.202
-1.465
0.576
1.650
1.620
- 0.415
- 0.050
0.523
0.776
- 0.145
0.0
-1.465
0.576
0.0
1.805
- 0.415
- 0.050
0.523
0.0
- 0.145
7.221
- 0.925
0.324
0.303
- 0.387
-0.100
0.324
0.0
0.967
- 0.282
- 0.130
0.303
0.0
-0.100
0.0
0.0
0.324
0.0
0.0
- 0.155
0.0
0.386
0.0
- 0.100
0.0
0.0
0.263
0.0
0. 0
- 0.321
0.0
0.236
0.0
- 0.170
3.444
- 0.226
0.303
- 5.370
1.188
- 0.114
- 0.110
0.277
- 1.783
- 0.024
0.0
- 0.226
0.303
0.0
0.784
- 0.114
- 0.110
0.277
0.0
- 0.024
0.0
0.0
0.303
0.0
0.0
0.038
0.0
0.366
0.0
- 0.024
0.0
0.0
0.576
0.0
0. 0
- 0.369
0.0
0.443
0.0
- 0.145
7.625
- 1.517
0.429
- 0.301
1.052
- 0.422
-0.230
0.249
0.587
- 0.200
0.0
- 1.517
0.429
0.0
0,802
- 0.422
-0.230
0.249
0.0
- 0.200
0.0
0.0
0.429
0.0
0.0
- 0.454
0.0
0.267
0.0
- 0.200
- 2.522
1.366
- 0.282
- 0.130
0.0
LiBr
- 1.319
14.262
- 1.319
0.0
0.248
3.271
1.387
0.248
0.0
1.516
0.0
0.0
0.248
0.0
0.0
- 0.412
- 0.412
- 0.348
0.015
0.301
2.639
-0.182
0.015
0.301
0.0
-0.182
0.0
0.285
0.0
- 0.182
LiI
NaF
14.108
0.0
- 1.222
- 1.222
0.192
4.427
1.097
0.192
0.0
1.019
-0.366
0.063
0.178
4.341
- 0.177
-0.366
0.063
0.178
0.0
- 0.177
0.0
0.0
0.192
0.0
0.0
- 0.298
0.0
0.171
0.0
- 0.177
NaC 1
12.527
0.0
0.0
- 1.367
- 1.367
0.512
3.016
1.232
- 0.376
- 0.010
0.374
1.820
- 0.150
0.512
0.0
1.467
- 0.376
- 0.010
0.374
0.0
- 0.150
0. 0
0.512
0.0
0.0
- 0.330
0.0
0.293
0.0
- 0.150
T A B L E I1 (continued).
Neighbor
L1 L2
L3
(+)
tx
tY
(+)
(-)
Ex
tz
NaB r
(Y
tz
x
[Y
(-)
tx
(z
NaI
5,
5,
TABLE I1 (continued).
Neighbor
Li L2 -53
(+)
Ex
Ey
(-)
Ez
5%
(-)
(+)
f,
tz
tx
EY
Ex
tz
RbI
KbBr
0
1
1
0
2
1
2
2
0
2
0
0
1
0
0
2
0
1
0
2
7.860
-1.517
0.417
0.729
0.825
-0.405
- 0.232
0.195
1.403
-0.209
0.0
-1.517
0.417
0.0
0.687
-0.405
- 0.232
0.195
0.0
-0.209
0.0
0.0
0.417
0.0
0.0
-0.456
0.0
0.188
0.0
-0.209
10.067
- 1.735
0.319
2.773
0.548
-0.488
-0.387
0.113
3.553
- 0.310
0.0
- 1.735
0.319
0.0
0.690
-0.488
-0.387
0.113
0.0
-0.310
0.0
0.0
0.319
0.0
0.0
- 0.546
0.0
0.082
0.0
-0.310
10.615
- 1.162
0.543
2.100
1.101
-0.293
0.021
0.345
1.478
- 0.097
0.0
- 1.162
0.543
0.0
1.166
- 0.293
0.021
0.345
0.0
- 0.097
0.0
0.0
0.543
0.0
0.0
0.247
0.0
0.276
0.0
- 0.097
-
CsF
1
1
1
2
1
0
1
1
0
2
1 1
2 2
2 2
3 0
2 2
0
0
1
0
0
2
0
1
0
2
52
RbCl
RbF
1
1
1
2
1
1
2
2
3
2
5,
8.614
- 0.690
0.534
- 0.491
1.295
- 0.174
0.092
0.397
- 0.033
0.0001
0.0
- 0.690
0.534
0.0
1.075
- 0.174
0.092
0.397
0.0
0.0001
0.0
0.0
0.534
0.0
0.0
- 0.056
0.0
0.376
0.0
0.0001
6.770
- 1.408
0.406
0.237
0.808
- 0.367
- 0.192
0.185
1.008
- 0.177
0.0
- 1.408
0.406
0.0
0.571
- 0.367
- 0.192
0.185
0.0
- 0.177
0.0
0.0
0.406
0.0
0.0
- 0.409
0.0
0.191
0.0
- 0.177
10.628
0.0
- 1.155
- 1.155
0.524
2.642
1.008
-0.289
0.025
0.314
1.802
- 0.102
0.524
0.0
1.117
-0.289
0.025
0.314
0.0
- 0.102
0.0
0.0
0.524
0.0
0.0
-0.237
0.0
0.242
0.0
- 0.102
3424
-3
A. M. KARO AND J . R . HARDY
propriate Fourier amplitudes for all the allowed
wave vectors after taking into account the effect of
first-neighbor relaxation on the short-range force.
The summation of the Fourier series requires
additional comment. Both the Fourier s e r i e s for
the displacement and the summation that determines
the relaxation energy have integrable singularities
in the long-wavelength limit. Throughout the calculations, we have used periodic boundary conditions imposed across the faces of a supercell having
the same symmetry a s the primitive unit cell of
the lattice. To see how the displacements and energies change with the sampling density and to determine whether the singularities at the Brillouinzone origin have introduced complications, we have
successively increased the number of points sampled on a uniform mesh within the first Brillouin
zone from 1000 to 8000 to 64 000 and, in several
cases, to 512 000. From this it is found that the
displacements converge rapidly for neighbors near
the defect. Even for more distant neighbors such
a s those listed in Table 111, the change in going
from 8000 to 64 000 i s only a few percent. Results
in Tables I1 and I11 for the displacements a r e for
the sampling density of 64 000 within the first zone.
Increasing the sampling density should not give appreciably different results. There a r e larger percentage changes in the relaxation energies calculated for sample densities of 8000 and 64 000, r e spectively, but the convergence i s very uniform.
Accurate extrapolation to the energy values corresponding to an infinitely dense sample of wave vectors i s readily possible. These extrapolated values
a r e those listed in Table I .
In Table I we also have the rcsults obtained by
Boswarva and ~ i d i a r d "using a refined Mott-Littleton theory. Although very close agreement exists
between our results and theirs, it would be unwise
to read any great significance into this since the
approximations in the two cases a r e very different.
For example, Boswarva and Lidiard used a more
highly developed model for the lattice energy, including repulsive interaction beyond first neighbors
and also including van der Waals attractions between
TABLE 111. x components of the relaxations a t lattice sites along three symmetry directions in the lattice. Values
a r e given in terms of percent of the near-neighbor distance in the perfect lattice and positive values indicate outward
displacements. The sign in the parentheses indicates the vacancy type. The [I001 direction i s taken to be equivalent to
the X direction.
Neighbor
L. Lg Lq
LiF
Ex(+)
NaCl
[,(-I
5,(+)
KBr
tx(-)
5,(+)
RbI
[,(-I
5,(+)
tJ-)
-3
CALCULATION O F P O I N T - D E F E C T ENERGIES
second neighbors.
IV. SUMMARY
The present results a r e very satisfactory and
represent the zero-order approximation. Subsequent extension of this initial investigation to more
refined lattice potential functions and to forces
evaluated at the relaxed positions has been carried
out1' for Frenkel-pair formation energies where
the relaxations a r e significantly larger. In modifying the zero-order approximation for E, t o allow
for relaxation of the first neighbors, the calculations become considerably more complex. How
ever, these refinements a r e essential for treating
an interstitial positive o r negative ion since the r e laxations about the defect a r e s o large that zeroorder results have little significance.
The lattice-statics method, then, avoids the difficulty inherent in the Mott-Littleton approach of
matching a discrete region I t o a continuum region
11. The two regions into which the lattice is divided
in the lattice-statics treatment describe the space
where the defect-lattice forces a r e corrected for
relaxation and the space where the forces a r e evaluated at the unrelaxed positions. From our present
knowledge, restricting the first region to first
neighbors of the defect seems to be sufficient. This
is readily checked by computing the displacements
and dipoles of various close neighbors. For other
than first neighbors, these a r e usually small.
Thus, the method of lattice statics offers a rigorous means of making self-consistent calculations
-
*Work performed under the auspices of the U. S.
Atomic Energy Commission.
'N. F. Mott and M. J. Littleton, Trans. Faraday Soc.
% 485 (1938).
'P. 0. LEwdin, A Theoretical Investigation into Some
fioperties of Ionic Crystals (Almqvist and Wiksells
boktryckeri AB, Uppsala, Sweden, 1948).
3 ~ W.
.
Flocken and J. R. Hardy, Phys. Rev. 175, 919
(1968); 177,1054 (1969).
h. H. Scholz, AERE Report No. R5449; Phys. Status
Solidi 2,
285 (1968).
825
'J. R. Hardy and A. B. Lidiard, Phil. Mag.
(1967).
...
3425
of charged-defect formation energies in ionic crystals. The next stage in such calculations, after
correcting for relaxation effects, would be to use
a more sophisticated form for the cohesive energy
of the perfect lattice, e. g., to include secondneighbor, short-range, and van der Waals t e r m s
a s done by Boswarva and ~ i d i a r d "in their MottLittleton calculations. The present results a r e
presented in the spirit of Mott and Littleton's original work: that is, t o demonstrate the validity of
the technique for a simple potential function. To
include all the refinements initially would be to
confuse and obscure the power and effectiveness of
this new technique. In comparison with experiment
the present results show the same tendency a s the
various Mott-Littleton results to be on the low side.
Although more refined calculations may prove bett e r in this respect, it should be noted that experimental values of E s a r e measured indirectly, depending significantly on theoretical analysis of the
experimental data, and a r e taken at several hundred
degrees centigrade, whereas the input parameters
generally available for the lattice-statics theory
a r e from experimental data at o r below room temperature.
ACKNOWLEDGMENT
We a r e indebted to I. W. Morrison of the Computation Division of the Lawrence Radiation Laboratory, University of California, for designing and
carrying through the computation reported in this
investigation.
6 ~ R.
.
Hardy, Phil. Mag. '7, 315 (1962).
'B. Szigeti, Proc. Roy. Soc. (London)
51 (1950).
8 ~ P.
. Ewald, Ann. Physik54, 519 (1917); 54, 557
(1917);
253 (1921).
9 ~ .M. Karo and J. R. Hardy, Lawrence Radiation
Laboratory Report No. UCRL-71475 (unpublished).
This report also contains detailed results pertaining to
both the temperature and the model dependence of the
lattice-statics method.
"I. M. Boswarva and A . B. Lidiard, Phil. Mag. g,
805 (1967).
"P. Schulze and J. R. Hardy, Bull. Am. Phys. Soc.
15, 51 (1970).
-
s,
m,